無年度 - 主題課程_線性映射：one to one and onto#107850

試卷資訊

選擇題 (1)

複選題

109中山電機
Consider the lincar mapping L: V→W. Let Ov and Ow be the zero vectors in V and W, respectively. Which of the following statements are true? (A) The condition L(v₁) = L(v₂) implies v = V2. (B) For any w W, there exists v V such that L(v) = w. (C) If L is onc-to-one, then L(v) = 0w implies v = 0v. (D) Ifv₁,v₂, , V_k are linearly independent, L(v₁), L(v₂), ...,L(v_k) are also linearly independent. (E)The condition c₁v₁+c2v₂+c_kv_k=0vimplies c₁L(v₁)+c₂L(v₂)+⋯+c_kL(v_k)=0w.

申論題 (5)

(a). Prove that if ker T= {0}; then T is one-to-one.(7 pts)

(b). Suppose T is one-to-one and {u₁,... ,u_k} is a linearly independent set of vectors in U. Prove that [T(u₁),....,T(u_k)] is a linearly independent set of vectors in V.(7 pts)

(c). Define UP(t)(a polynomil of degree 2 and its standard form is P(t) = a₀+ a₁t + a₂t²),V R³, and T(U) = Find U such that the image under T of Uis [11, 1, -1]^T. (7 pts)

(a) (5%) Find a vector u = (u_x, u_y, u_z) such that T(u) - u and.

(b) (5%) Is T : R₃ → R₃ one-to-one and onto? Why or why not?